4.3 INSURANCE DEMAND IN THE KAHNEMAN-TVERSKY MODEL (PROSPECT THEORY, PT)

Psychologists have observed that people often assess their economic situation based not on their absolute income, but rather on the change therein or on deviations from their starting position.^{[1]} Kahneman and Tversky's prospect theory takes this aspect of the human psyche into consideration.^{[2]} In this theory the utility function:

a) is strictly convex in the region of losses relative to a reference point (inflection point R in Figure 20 below) and strictly concave in the region of gains relative to the same reference point. This corresponds to an assumption of decreasing sensitivity to a stimulus as the stimulus gets more intense, i.e. an assumption of decreasing marginal utility from a growing gain and a decreasing marginal loss as the loss increases;^{[3]}

b) is steeper in the loss region than in the gain region, i.e. the decrease in utility given a loss of a unit of income is greater than the increase in utility given a gain of a unit of income. This is consistent with the fact that decision-takers assign a greater weight to a loss than to an equal gain.^{[4]}

What we have, then, is an asymmetric utility function that displays risk aversion in the region of gains and risk attraction in the region of losses relative to the reference point, which is the inflection point of the function.^{[5]}

The Kahneman-Tversky utility function in Figure 20 is consistent with this— a nominally smaller loss has a greater effect on the agent's utility than a nominally larger gain in the agent's wealth, i.e. even though .

Figure 20: The Kahneman-Tversky utility function

The degree of convexity/concavity decreases with distance from the reference point R. This is in line with psychological findings^{[6]} according to which the impact on the human psyche of a unit marginal increase in a stimulus declines as the strength of the stimulus increases.

Besides the utility function, a key factor in Kahneman and Tversky's theory for choice under uncertainty is the conversion of objective probabilities into subjective ones. Economic psychology has demonstrated that people tend to overweight the probability of extremely unlikely events and underweight the probability of normal phenomena. Prospect theory captures this using a weighting function that converts the objective probability of a phenomenon p into a subjective probability, where δ Î (0;1) is the parameter determining the degree of overweighting of the probability.^{[7]}

For example, for an agent with property d who is able to buy insurance at a cost of a against a loss of L with a probability of p, the key relation in Kahneman and Tversky's prospect theory is^{[8]}

This value function in Kahneman-Tversky prospect theory is therefore a modified utility function. For "sensible" values of δ (i.e. for values consistent with the degree of overweighting of insurance loss probabilities in the hundreds of per cent at most), the function is still convex-concave and still steeper in the loss region than in the gain region.

The higher sensitivity to losses than to gains implies that a "fair” game [for instance tossing a coin, where both the magnitude and probability of gain and loss are equal] is acceptable to our model agent — he does not perceive it to be detrimental. This attitude to risk influences his insurance choice — a potential insurance loss of b money units has a larger weight in his decision than a potential increase in wealth of a money units saved on the premium and he is willing to buy insurance at a premium substantially higher than the expected loss. For such an agent, it is not profitable to choose coinsurance at a reduced premium, whereas in the case of the standard strictly concave utility function such coinsurance would be seen as worthwhile.^{[9]} Using the Kahneman-Tversky value function we can also explain the economically irrational tendency to take sunk costs into account when making decisions^{[10]} — out-of-pocket expenditure (albeit in the past) is regarded as a loss, whereas opportunity costs are seen as (less valued) for egone profits.

The Kahneman-Tversky value function is sensitive to the ranking of alternatives. Pooled profits and losses are assessed more favourably than profits and losses considered separately. Even merely changing the sequence in which problems are solved changes the assessment.

Figure 21: The Kahneman-Tversky value function for a very poor agent — such an agent will not take out insurance

Figure 21 shows the insurance threshold for the Kahneman-Tversky value function in prospect theory. There is a fundamental difference here from the analogous function in the previous section: people above the threshold take out insurance. Poorer agents have negative risk aversion (a strictly convex utility function) and are therefore risk seekers, i.e. they do not eliminate risk with insurance. The threat (with a 1:5 probability) of a loss of L = 250 money units corresponds to point M, which is one-fifth (since p = 20%) of the way along AB, closer to d. This point lies above N = [d - a, u (d - a)] for every sensible premium, i.e. For every premium that is higher than the expected loss E(L) and lower than the potential loss L (in our case for 50 ≤ a < 250). Any agent below the point of inflection of the Kahneman-Tversky value function in prospect theory will reject insurance because he is risk-attracted.

An agent whose income will — even after paying the premium — remain above the point of inflection of the Kahneman-Tversky value function but who would be driven below this point by an insurance loss (into the negative risk aversion region] can be described as "middle class”. Such an agent is depicted in Figure 22.

Figure 22: The Kahneman-Tversky value function for a middle-class agent — such an agent will take out insurance

For wealthy agents the situation is the same as in the previous section: given extremely high income (above the threshold] insurance ceases to be worthwhile.

The attitude to insurance of an agent with a Kahneman-Tversky utility function therefore differs from that of an agent maximizing the expected utility of wealth primarily in the case of agents with below-average income. In the previous model the poor chose insurance, whereas as here they do not. As for wealthy agents, their strategy differs from the previous model only in that the wealth threshold above which agents take out insurance can increase. This will happen if the threshold is just above the point of inflection of the Kahneman- Tversky curve.

The insurance demand function here is significantly less price elastic compared to the expected utility of wealth maximization approach. A poor agent is a risk-seeker (has negative risk aversion] and so does not react to a reduction in the premium. The threshold above which a wealthy agent will take out insurance is determined more by the position of the reference (inflection] point than by the premium amount.

The insurance market in the Kahneman-Tversky model has lower and less price elastic demand than the previous model. Conversely, the income elasticity of demand is higher for the Kahneman-Tversky asymmetric value function, because as income increases some agents move into the category of wealthy agents, who, unlike poorer ones, are risk averse. The Kahneman-Tversky value function therefore exhibits particularly high income elasticity in the vicinity of its point of inflection.

In the expected utility of wealth maximization model we observed a paradoxical view of insurance among the poorest (the poor have more insurance the poorer they are), whereas the Kahneman-Tversky approach succeeds in capturing the aversion to insurance that exists among the poor in reality. However, the motive — negative risk aversion — is debatable. This trait may exist in the poorer classes with regard to gambling, for example, but it is more than debatable in the case of insurance. Moreover, in the Kahneman-Tversky model even the middle classes do not buy insurance, whereas in reality they account for most of the demand for insurance. Insurance is taken out by agents lying above the threshold d≥d2 and below the threshold d≤d1 for which point [a, d) lies in region Q in the following figure.

Figure 23: Wealth thresholds d1 and d2 in the Kahneman-Tversky value function model versus insurance premium a.

Related to this is the insurance demand function, i.e. the relation between demand and price (premium a):

Figure 24: The insurance demand function in the Kahneman-Tversky value function model with uniform income distribution

In the following section we will compare the insurance market demand curves for all three models studied (see Figures 13, 18, and 24).

[1] The aforementioned Weber–Fechner law from psychology—see section 1.3.2.

[2] See Tversky, A., Kahneman, D.: The Framing of Decisions and the Psychology of Choice. Science 211, 4481(1981): 453–58, and Tversky, A., Kahneman, D.: Judgment under Uncertainty: Heuristics and Biases. Science 185, 4157(1974): 1124–31.

[3] In the later cumulative prospect theory the assumptions of a convex-concave utility function are modified in the sense that the concavity changes into convexity and vice versa when there is a very low probability of an extreme outcome. See Skořepa, M.: Daniel Kahneman a psychologické základy ekonomie. Politická ekonomie 52, 2(2004): 247–55.

[4] Kahneman and Tversky derived the shape of the utility function from empirical findings, according to which the steepness of the utility function in the loss region is around 2.2 times greater than in the gain region. For comparison, in section 4.4, where we compare the EU model (section 4.2), the Kahneman–Tversky prospect theory (section 4.3) and the Pareto survival probability maximization model (section 4.1), we use the same utility function in the gain region as in the previous section, i.e. for d ≥ 0, and in the loss region we assume a constant aversion to loss relative to gain: u(d) = –2.2 • u(–d) for d < 0.

[5] In a later version of their theory (known as cumulative prospect theory, CPT) the authors responded to new empirical findings by making the shape of the value function dependent on the magnitude of the probability of the gain or loss and differentiating between the risk relationships for normal deviations and extreme deviations from the reference point.

[6] See Frank, R. H.: Microeconomics and Behavior. New York: McGraw-Hill, 2006, chapter 8, 259–83.

[7] Kahneman and Tversky again derived the shape of the weighting function π(p) from empirical findings. For the following comparison we use a value of δ = 0.7, at which the overweighting of a phenomenon with a probability of 1% is roughly fourfold (π(0.7) = 3.8). This value may be appropriate for the decision to buy insurance (where the loss probability is in whole percentage numbers), but is not so for the decision to buy a lottery ticket (where the probability is several orders of magnitude lower).

[8] Likewise, for the decision to buy a lottery ticket at price q with payoff V with probability p, the key relation for the Kahneman–Tversky utility function is π(p) • u(d + V) ~ [1 – π(p)] • u(d – q).

[9] See Frank, R. H.: Microeconomics and Behavior. New York: McGraw-Hill, 1994, chapter 8, 259–83.

[10] Sunk costs are costs that have already been incurred and cannot be affected by the agent's current choice. He should therefore not take them into account when making his decision.

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